Kinetic Theory of Gases

Transcription

1 Kinetic Theory of Gases Physics 1425 Lecture 31 Michael Fowler, UVa

2 Bernoulli s Picture Daniel Bernoulli, in 1738, was the first to understand air pressure in terms of molecules he visualized them shooting around very rapidly in a closed container, supporting a weight as shown by constantly bouncing off the underside of the piston. Given more room, they would rush in to fill the new space, just as a gas is observed to do. No-one believed him. Applet here.

3 One Dimensional, One Molecule Gas The molecule roundtrips in time 2L/v, so it bounces off the piston v/2l times per sec, each time delivering momentum 2mv, so the piston will pick up momentum from this gas at rate 2mv x v/2l per second. Force from gas on piston: F = rate of change of momentum = mv 2 /L. An equal opposite force must be supplied from outside to keep the piston at rest. Animation! V v 1-D gas: molecule bounces between ends of cylinder. L

4 Molecule in a Two-Dimensional Box Assume perfectly elastic collisions with all walls. The molecule will follow a zigzag path, the time between collisions with the same end, say the end at x = L, is now 2L/v x, and the momentum transferred per collision is 2mv x, so the average force on the end is mv x2 /L. This will still hold good in three dimensions. a y v y v x 0 x L

5 N Molecules in an L x L x L Cube Assume first that we have a very large number N of molecules bouncing around, so small that they don t hit each other, each follows its own zigzag path. The force on the right-hand wall at x = L is just the sum of the forces from each one, so F = mv x12 /L +mv x22 /L +mv x32 /L + + mv xn2 /L.

9 Finding the Ideal Gas Law ( ) We ve established that PV = 3N 2mv and we know that for very weakly interacting gases, PV = nrt. These two equations must be the same! The equivalence is most simply expressed using Boltzmann s constant, k = R/N A (= 1.38 x J/K). PV =nrt = nn A kt = NkT, so mv = kt Absolute temperature is proportional to average molecular kinetic energy.

10 Average Speed of Air Molecules Maxwell and co were very smart guys they figured out accurately the average speed of air molecules before they had any idea how big the molecules were! They just used ( ) PV = N mv = Nmv = M v where M is the total mass of the gas in the box.

11 Average Speed of Air Molecules 1 2 Let s see what PV = 3 M v gives for the speed of air molecules in this room (we are, of course, averaging here over oxygen and nitrogen plus a tiny amount of other stuff). Let s take a one meter cube: it will contain about 1.3 kg of air. The pressure P = 10 5 N/m 2, close enough, so PV = 10 = 3 (1.3) v 2 giving the root mean square value v = 480 m/s.

12 Clicker Question Since oxygen, nitrogen and helium all satisfy the same gas law PV = nrt at room temperature, we conclude that: A. All have the same (rms) root mean square average molecular speed B. All have the same average molecular kinetic energy C. Neither of the above is true.

13 Clicker Question If we take the average speed of oxygen molecules in this room to be 480 m/s, what would be the average speed of helium atoms that leaked from a balloon into the room? A. 480 m/s B. 960 m/s C m/s D m/s

14 The Speed Distribution Although the molecules fly freely almost all of the time, they do collide occasionally. Assuming random elastic collisions, there will be transfer of energy, typically of order kt, from one to another in a collision. The chances of a particular molecule picking up kt n times in a row is similar to the chances of a coin toss coming up heads n times in a row. that is, high energies are exponentially unlikely.

15 Maxwell s Speed Distribution Maxwell did the math precisely, and found the probability of a molecule having a high energy at a given moment did drop exponentially with energy: Probability (speed = v) e 1 2 mv / 2 kt meaning that for each extra kt of energy, the probability of finding a particle with that energy drops by 1/e 0.37 more than a factor of 2, because the average amount picked up per collision is less that kt. 1 2 mv / kt 2 (Maxwell s exact result is f v = 4π N ve.) 2π ( ) m kt 3 2 2

16 Escaping from a Planet Maxwell s speed distribution makes it quite easy to predict which gases can escape from planetary atmospheres. For the Earth, v escape = 11 km/sec. The upper atmosphere has parts as hot as 1000K. 2 mv kt 3 escape /2 The speed distribution includes e = e where we ve used 2mv = 2kT. For H 2, at 1000K, the fraction of molecules at escape velocity is of order 10-6, for He 10-12, for O This means the H 2 will escape almost instantly, the He pretty quickly, and the O 2 never. 2 2 /2 v v

17 Clicker Question Which of the following gases would you expect to be dominant in the Martian atmosphere? A. H 2 B. He C. N 2 D. H 2 0 E. CO 2

18 Real Gases (This is of course chemistry.) Water has the three phases as shown on this pressure/ temperature graph. They meet at the triple point a definite P and T, useful as a reference point in fixing temperatures. Freezing and boiling points vary with pressure. At the critical point, liquid and vapor become the same. The solid-liquid dividing line slopes forwards for almost all substances water is an exception.

19 Vapor Pressure and Humidity The H 2 O molecules in liquid water strongly attract each other, holding the liquid together. But these molecules are still jiggling around, with a Maxwell speed distribution. This means a fraction of them near the surface are moving fast enough to escape, forming a vapor above the surface. In a closed container, with enough water present, an equilibrium situation is reached between escaping and returning molecules.

20 Water and Vapor in Equilibrium In equilibrium in a closed container, the molecules in the vapor have the same average kinetic energy as the air molecules, so exert pressure on the walls of the container proportionate to their numbers. This is the saturated vapor pressure. It varies with temperature like e a/t, not surprising since its origin is molecules fast enough to escape. a Air + water vapor Water

21 Water Vapor Pressure At room temperature, saturated vapor pressure is about 2.5% of atmospheric pressure. At 100 C, it equals atmospheric pressure: this means small bubbles formed in the liquid by fast moving molecules coming together are no longer crushed by the surrounding atmospheric pressure, the water boils. In mountain resorts like Aspen, water boils at a lower temperature, producing inferior tea.

22 Relative Humidity and Dew Point If water is constantly boiled off in a closed room, it is not possible to establish an equilibrium situation with vapor pressure above the saturated value for that temperature further water will condense out on the walls, etc. At this point, relative humidity = 100%. Relative humidity = vapor pressure/saturated vapor pressure Dew point: temperature at which dew forms that is, water condenses out as the air cools.

Data Sheet Extract The theory for ideal gases makes the following assumptions: The gas consists of very small atoms or molecules (spoken of as particles), each of which has an identical mass and are perfectly

Name Period Gas Laws Kinetic energy is the energy of motion of molecules. Gas state of matter made up of tiny particles (atoms or molecules). Each atom or molecule is very far from other atoms or molecules.

Kinetic Theory of Gases Important Points:. Assumptions: a) Every gas consists of extremely small particles called molecules. b) The molecules of a gas are identical, spherical, rigid and perfectly elastic

Gas Laws and Kinetic Molecular Theory The Gas Laws are based on experiments, and they describe how a gas behaves under certain conditions. However, Gas Laws do not attempt to explain the behavior of gases.

OpenStax-CNX module: m42217 1 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons

UNIT HEAT. KINETIC THEORY OF GASES.. Introduction Molecules have a diameter of the order of Å and the distance between them in a gas is 0 Å while the interaction distance in solids is very small. R. Clausius

CLASSICAL CONCEPT REVIEW 8 Kinetic Theory Information concerning the initial motions of each of the atoms of macroscopic systems is not accessible, nor do we have the computational capability even with

Chemistry 13: States of Matter Name: Period: Date: Chemistry Content Standard: Gases and Their Properties The kinetic molecular theory describes the motion of atoms and molecules and explains the properties

Chapter 14 he Ideal Gas Law and Kinetic heory Chapter 14 HE IDEAL GAS LAW AND KINEIC HEORY REIEW Kinetic molecular theory involves the study of matter, particularly gases, as very small particles in constant

Chapter 19: THE KINETIC THEORY OF GASES 1. Evidence that a gas consists mostly of empty space is the fact that: A. the density of a gas becomes much greater when it is liquefied B. gases exert pressure

5. The Kinetic Theory of Gases Introduction and Summary Previously the ideal gas law was discussed from an experimental point of view. The relationship between pressure, density, and temperature was later

The first scheduled quiz will be given next Tuesday during Lecture. It will last 5 minutes. Bring pencil, calculator, and your book. The coverage will be pp 364-44, i.e. Sections 0.0 through.4. 0.7 Theory

Boltzmann Distribution Law The motion of molecules is extremely chaotic Any individual molecule is colliding with others at an enormous rate Typically at a rate of a billion times per second We introduce

Kinetic Molecular Theory This explains the Ideal Gas Pressure olume and Temperature behavior It s based on following ideas:. Any ordinary sized or macroscopic sample of gas contains large number of molecules.

Kinetic Molecular Theory Particle volume - The volume of an individual gas particle is small compaired to that of its container. Therefore, gas particles are considered to have mass, but no volume. There

j1 1 Introduction The aim of this book is to provide an understanding of the basic processes, at the atomic or molecular level, which are responsible for kinetic processes at the microscopic and macroscopic

The Gas Laws Describe HOW gases behave. Can be predicted by the theory. The Kinetic Theory Amount of change can be calculated with mathematical equations. The effect of adding gas. When we blow up a balloon

The Kinetic Theory of Gases Sections Covered in the Text: Chapter 18 In Note 15 we reviewed macroscopic properties of matter, in particular, temperature and pressure. Here we see how the temperature and

KINETIC THEORY OF GASES Boyle s Law: At constant temperature volume of given mass of gas is inversely proportional to its pressure. Charle s Law: At constant pressure volume of a given mass of gas is directly

Name: Class: _ Date: _ ID: A Chapter 3 Assessment Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which state of matter has a definite volume but a variable

EXPERIMENT 15: Ideal Gas Law: Molecular Weight of a Vapor Purpose: In this experiment you will use the ideal gas law to calculate the molecular weight of a volatile liquid compound by measuring the mass,

SESSION 7: KINETIC THEORY OF GASES Key Concepts In this session we will focus on summarising what you need to know about: Kinetic molecular theory Pressure, volume and temperature relationships Properties

1 P a g e Physics Notes Class 11 CHAPTER 13 KINETIC THEORY Assumptions of Kinetic Theory of Gases 1. Every gas consists of extremely small particles known as molecules. The molecules of a given gas are

Humidity, Evaporation, and Boiling Bởi: OpenStaxCollege Dew drops like these, on a banana leaf photographed just after sunrise, form when the air temperature drops to or below the dew point. At the dew

1 of 6 Thermodynamics Summer 2006 Kinetic Theory & Ideal Gas The study of thermodynamics usually starts with the concepts of temperature and heat, and most people feel that the temperature of an object

Chapter 10 - Gases Gas - a substance that is characterized by widely separated molecules in rapid motion. Mixtures of gases are uniform. Gases will expand to fill containers (compare with solids and liquids

The Ideal Gas Law By: OpenStax College Online: This module is copyrig hted by Rice University. It is licensed under the Creative Commons Attribution License: http://creativecommons.org/licenses/by/3.0/

Thermodynamics AP Physics B Name Multiple Choice Questions 1. What is the name of the following statement: When two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium

Chapter 13 The skunk releases its spray! Within seconds you smell that all-too-familiar foul odor. You will discover some general characteristics of gases that help explain how odors travel through the

Bellringer You are already familiar with the most common states of matter: solid, liquid, and gas. For example you can see solid ice and liquid water. You cannot see water vapor, but you can feel it in

What is a state? Set of values that describe the current condition of a system, usually in equilibrium Is this physics different than what we have learned? Why do we learn it? How do we make the connection

PROPERTIES OF GASES or GAS LAWS 1 General Properties of Gases There is a lot of empty space in a gas. Gases can be expanded infinitely. Gases fill containers uniformly and completely. Gases diffuse and

CHATER 3. The atmosphere is a homogeneous mixture (a solution) of gases.. Solids and liquids have essentially fixed volumes and are not able to be compressed easily. have volumes that depend on their conditions,

Temperature, Expansion, Ideal Gas Law Physics 1425 Lecture 30 Michael Fowler, UVa Everything s Made of Atoms This idea was only fully accepted about 100 years ago in part because of Einstein s analysis

Describe the strength of attractive forces between particles. Describe the amount of space between particles. Can the particles in this state be compressed? Do the particles in this state have a definite

Page 1 Unit 3: States of Matter Practice Exam Multiple Choice. Identify the choice that best completes the statement or answers the question. 1. Two gases with unequal masses are injected into opposite

1.4.6-1.4.8 Gas Laws Heat and Temperature Often the concepts of heat and temperature are thought to be the same, but they are not. Perhaps the reason the two are incorrectly thought to be the same is because

CHAPTER 3 PROPERTIES OF NATURAL GASES The behavior of natural gas, whether pure methane or a mixture of volatile hydrocarbons and the nonhydrocarbons nitrogen, carbon dioxide, and hydrogen sulfide, must

Chapter 17 Temperature, Thermal Expansion, and the Ideal Gas Law Units of Chapter 17 Atomic Theory of Matter Temperature and Thermometers Thermal Equilibrium and the Zeroth Law of Thermodynamics Thermal

Note: numbers used in solution steps are different from your WebAssign values. Page 1 of 8 Note: numbers used in solution steps are different from your WebAssign values. 1. Walker3 17.P.003. [565748] Show

Kinetic Theory of Gases Kinetic Theory of Gases Chapter 33 Kinetic theory of gases envisions gases as a collection of atoms or molecules. Atoms or molecules are considered as particles. This is based on

Chapter 4 The Properties of Gases Significant Figure Convention At least one extra significant figure is displayed in all intermediate calculations. The final answer is expressed with the correct number

MISN-0-157 TEMPERATURE AND PRESSURE OF AN IDEAL GAS: THE EQUATION OF STATE TEMPERATURE AND PRESSURE OF AN IDEAL GAS: THE EQUATION OF STATE by William C. Lane Michigan State University 1. Introduction a.

13 STATES OF MATTER SECTION 13.1 THE NATURE OF GASES (pages 385 389) This section introduces the kinetic theory and describes how it applies to gases. It defines gas pressure and explains how temperature

Lecture 9 State of gas described by (n,p,v,t) n # moles P pressure V volume T (absolute) temperature (K) Sample Problem A balloon filled with helium has a volume of 1.60 L at 1.00 atm and 25oC. What will

Chapter 18. The Micro/Macro Connection Heating the air in a hot-air balloon increases the thermal energy of the air molecules. This causes the gas to expand, lowering its density and allowing the balloon

Chapter 7 Ideal and Real Gases Gas, Liquid, and Solid Chemical Process calculation III Gas: a substance in a form like air, relatively low in density and viscosity Liquid: a substance that flows freely

Define linear momentum (and appreciate the vector nature of momentum) net force on a body impulse of a force a perfectly elastic collision an inelastic collision the radian gravitational field strength

Topic 3b: Kinetic Theory What is temperature? We have developed some statistical language to simplify describing measurements on physical systems. When we measure the temperature of a system, what underlying

TEACHER BACKGROUND INFORMATION THERMAL ENERGY In general, when an object performs work on another object, it does not transfer all of its energy to that object. Some of the energy is lost as heat due to

PHYS-2010: General Physics I Course Lecture Notes Section XIII Dr. Donald G. Luttermoser East Tennessee State University Edition 2.5 Abstract These class notes are designed for use of the instructor and

Chapter 2 Student Reading Atoms and molecules are in motion We warm things up and cool things down all the time, but we usually don t think much about what s really happening. If you put a room-temperature

Episode 603: Kinetic model of an ideal gas This episode relates the gas laws to the behaviour of the particles of a gas. Summary Discussion and demonstration: explaining pressure in terms of particles.

Version 001 HW03-Non Ideal, Gas Mixtures & KMT vandenbout (52130) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page find all choices before answering.

I. MOLECULES IN MOTION: A. Kinetic Molecular Theory (KMT) = the idea that particles of matter are always in motion and that this motion has consequences. 1) theory developed in the late 19 th century to

Physics A Unit: G484: The Newtonian World 1(a) State Newton s second law of motion. The resultant force on an object is proportional to the rate of change of momentum of the object In part (a) the candidate

6 N08/4/PHYSI/SP2/ENG/TZ0/XX+ A2. This question is about ideal gases. (a) State what is meant by an ideal gas....... For an ideal gas define internal energy. state and explain how the internal energy and

The particulate nature of matter Solids, liquids and gases The kinetic theory of matter Explaining the states of matter Changes of state An unusual state of matter An unusual change of state Heating and

Name: Class: Date: Atmospheric Properties Short Study Guide Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. Earth s atmosphere contains more

A cart full of water travels horizontally on a frictionless track with initial velocity v. As shown in the diagram, in the back wall of the cart there is a small opening near the bottom of the wall that